Spin Measurements of Accreting Black Holes: A Foundation for X-Ray Continuum Fitting

Transcription

1 Spin Measurements of Accreting Black Holes: A Foundation for X-Ray Continuum Fitting The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Steiner, James Spin Measurements of Accreting Black Holes: A Foundation for X-Ray Continuum Fitting. Doctoral dissertation, Harvard University. Citable link Terms of Use This article was downloaded from Harvard University s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#laa

2 Spin Measurements of Accreting Black Holes: A Foundation for X-ray Continuum Fitting A dissertation presented by James Francis Steiner to The Department of Astronomy in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Astronomy Harvard University Cambridge, Massachusetts March 2012

3 c 2012 James Francis Steiner All rights reserved.

4 iii Thesis Advisor: Doctor Jeffrey E. McClintock James Francis Steiner Spin Measurements of Accreting Black Holes: A Foundation for X-ray Continuum Fitting Abstract Remarkably, an astrophysical black hole has only two attributes: its mass and its spin angular momentum. Spin is often associated with the exotic behavior that black holes manifest such as the production of relativistic and energetic jets. In this thesis, we advance one of the two primary methods of measuring black hole spin, namely, the continuum-fitting method by (1) improving the methodology; (2) testing two foundational assumptions; and (3) measuring the spins of two stellar-mass black holes in X-ray binary systems. Methodology: We present an empirical model of Comptonization that self-consistently generates a hard power-law component by upscattering thermal accretion disk photons as they traverse a hot corona. We show that this model enables reliable measurements of spin for far more X-ray spectral data and for more sources than previously thought possible. Testing the foundations: First, by an exhaustive study of the X-ray spectra of LMC X 3, we show that the inner radius of its accretion disk is constant over decades and unaffected by source variability. Identifying this fixed inner radius with the radius of the innermost stable circular orbit in general relativity, our findings establish a firm foundation for the measurement of black hole spin. Secondly, we test the customary assumption that the inclination angles of the black-hole s spin axis and the binary s orbital axis are the same; for XTE J we show that they are aligned to within 12 by modeling the kinematics of the large-scale jets of this microquasar. Measuring spins: We have made the first accurate continuum-fitting spin measurements of the black hole primaries in H and XTE J For this latter black hole, we have also measured its spin using the other leading method, namely, modeling the broad red wing of the Fe Kα line. As we show, these two independent measurements of spin are in agreement.

5 Contents Abstract Acknowledgments Dedication iii ix xiii 1 Introduction Continuum-Fitting Measurements of Spin Spectral Model Foundations The ISCO Spin-Orbit Alignment New Spin Measurements H XTE J : A Joint CF and Fe Kα Study A Simple Comptonization Model Introduction The Model: simpl Green s Functions simpl-2: simpl Comparison to comptt iv

6 CONTENTS v Bulk Motion Comptonization Data Analysis Steep Power Law State Thermal Dominant State Comparison of simpl and powerlaw Discussion Summary Appendix: XSPEC Implementation Measuring Black Hole Spin via the X-ray Continuum Fitting Method: Beyond the Thermal Dominant State Introduction Observations & Analysis Results Final Selection of the Data via the Scattered Fraction Comparison with Other Comptonization Models Dependence on the Dynamical Model Discussion The Constant Inner-Disk Radius of LMC X 3: A Basis for Measuring Black Hole Spin Introduction Observations Flux Calibration Analysis Data Selection Results Discussion

7 CONTENTS vi 5 Modeling the Jet Kinematics of the Black Hole Microquasar XTE J : A Constraint on Spin-Orbit Alignment Introduction Data The Jet Model Markov Chain Monte Carlo MCMC in Practice Results Two Preliminary Models Our Adopted Model Constraining Spin-Orbit Alignment Radio Intensities and Asymmetric-Jet Models Discussion Conclusions The Distance, Inclination, and Spin of the Black Hole Microquasar H Introduction Data The Ballistic Jets: Model and Results X-ray Continuum-Fitting Analysis Conclusions The Spin of the Black Hole Microqusar XTE J via the Continuum-Fitting and Fe-Line Methods Introduction Observations Continuum-Fitting Analysis RXTE Data Selection

8 CONTENTS vii Results I: Continuum Fitting using smedge Continuum Fitting: Towards a Self-Consistent Disc + Reflection Model A Variant of the Power-Law Model simpl Results II: Continuum Fitting using ireflect and reflionx Continuum Fitting: Error Analysis and Final Spin Result Spin from Reflection Features Phenomenological Models ASCA Reflection Analysis ASCA Spin from reflection features RXTE Discussion A Combined Fe Kα and CF Result Testing the No-Hair Theorem Confronting GRMHD Simulations The Question of Alignment Implications of a Low-Spin Microquasar Conclusion Appendix: Continuum-Fitting: Assessing the Systematic Uncertainties Model Parameters Model Components Flux Black-Hole Mass, Inclination and Distance Rolling Together the Uncertainties Conclusions and Future Directions Summary: Spectral Model Summary: Foundations Summary: New Spin Measurements

9 CONTENTS viii 8.4 New Horizons for Black Hole Spin References 207

10 Acknowledgments What a remarkable feeling, to have a completed dissertation behind me (presumably in front of you). While the feeling is so very sweet, it is sad to be at a fork in the road which will lead me away from what has been a wonderful chapter in my life, and away from many friends and colleagues. Before we go diving into the world of black holes, the fortunate but impossible task of thanking the many people who helped me to obtain this PhD is at hand. There are a great many friends, teachers, and mentors who have been integral in this. While this in no way adequately matches the depth or reach of my thanks, I hope to at least pay tribute to some of the major culprits here. First and foremost, I must thank my wonderful wife Christina. She is more wonderful than laughter. Christina, I can t imagine any of this without you. Thank you to all of my family. Mom, Dad, Katherine, Adam, Mike, Grandma and Grandpa, Stan, Deb, Vanessa, Elizabeth, and Uncle Jim, I love you all. You inspire the best in me. The best part of grad school for me has been the wonderful set of friendships I have formed. Jeff, you re a major part of this too, but I ll come back to you later. Thank you Gurtina Besla, Sarah Ballard, Robert Marcus and Robert Harris, Sasha Tchekhovskoy, Dave Musicant, Manasvita Joshi and Karthikeyan Karunanidhi, Kelly and Dustin Sorel, Laura Blecha, Ryan Hickox, Wen-fai Fong, Lauranne Lanz, Sui Ann Mao, Chris Hayward, Sumin Tang, Manuel Torres, Laura Brenneman, Dan Castro, and Diego Munoz. I m especially grateful to Joey Neilsen, my office-mate, sounding board, and close friend. Aside from this group of Bostonians, I also want to thank Steve and Justin, Swati, Tomomi, Nastaran, Mangala, Joel, Toby, and Renee. ix

11 CONTENTS x I couldn t have asked for a better crew of friends. Thanks to my many teachers and mentors throughout the years: Brian McNamara, Valerie Young, Michael Moore, Saw Hla; all of you showed great patience with a noisy undergraduate who had too many ideas and too little knowledge to sort them out. Larry Curtis, you and Maj were such a delight. I learned a tremendous amount from you, and the paper we wrote together (guided very selflessly by you) was a landmark for me. Thank you so much Larry. The Honors Tutorial College at Ohio University was a fantastic program and gave great preparation for graduate school and offered a wealth of opportunity to pursue research. Ann Fidler, Joe Shields, and Ann Brown were a great part of making such a good program for students in the department. But I want to single out Tom Statler, whose grit and commitment to his students and to scientific ethos is truly impressive. I am lucky to have grown up attending a wonderful public school system. Thank you to the teachers at Maumee City Schools, including Mr. Dreyfus, Mme Greenberg, Mr. Dick, Mr. Tadsen, Mrs. Healey, Mrs. Leach, Mr. Lowry, and especially Dave Jerman, a wonderful teacher and now a close friend. Since I started working with Jeff, I have been fortunate to collaborate closely with a wonderful group of people. Lijun Gou, Ron Remillard, and Ramesh Narayan have been absolutely crucial to the work presented in this thesis. Lijun s tireless work ethic is impressive, and his always open door and ready ear have made collaborating together a real pleasure. Ron s tremendous ability with data is matched only by Ramesh s boundless knowledge of theory and scientific intuition. Ramesh s weekly group meetings with students and postdocs including Sasha Tchekovskoy, Bob Penna, Yucong Zhu, Jifeng Liu, C-K Chan, Akshay Kulkarni, Allison Farmer, Jon McKinney, Rebecca Shafee, Ryan Hickox, and others has been a highlight of my

12 CONTENTS xi time at CfA. I am very grateful to the Astronomy Department at Harvard for making the CfA a wonderful place for a student to thrive. Thanks to Jean Collins, Donna Adams, Donna Wyatt, Charles Alcock, and Jim Moran for their support from on high. Peg Herlihy has been a godsend, and helped out in every way, especially this last year as Christina and I moved away and managed working remotely. Peg s moral and academic support has gone far above and beyond. My thanks to Irwin Shapiro, Avi Loeb, Doug Finkbeiner, Josh Grindlay, Julia Lee, Dave Charbonneau, Alyssa Goodman, and the faculty and scientists at CfA. Lastly, I offer these words about my dear friend and advisor Jeff McClintock. Jeff s all around knowledge of X-ray science and every aspect of black hole behavior is truly remarkable. But more powerful still is his ability to inspire in others his enthusiasm and love for research, for black holes, and for science. He has been a bastion of support and wisdom throughout my time at CfA. Jeff and I have spent countless hours enthralled in the wonder of our black holes (and many other hours I treasure just as much exploring the tributaries of the mind). Jeff is unbelievably dedicated to the people and ideas he holds dear. Although he has always been overly busy in our more than five years of working together, Jeff s door has always been open. It wouldn t be right to say Jeff made time for us to meet, since that would convey completely the wrong sense of things. The time spent in our meetings was clearly a joy for Jeff, and that is one small but beautiful and telling aspect of what working with Jeff has been like. I could never have asked for or expected to have a mentor like Jeff. I am unbelievably fortunate. Jeff, I know we will continue to work together closely, and that knowledge eases the sting of how tremendously much I will miss spending my time with you.

13 CONTENTS xii Now, as Jeff will appreciate more than anyone, on to the science.

14 For Jeff xiii

15 Chapter 1 Introduction Adapted from J. F. Steiner, J. E. McClintock, R. Narayan, L. Gou Proceedings of Science, HTRS 2011, 2011, 019. Astrophysical black holes are among the most important objects in the world of modern physics. They inhabit the unknown nexus between quantum mechanics and Einstein s relativity. At the same time, a black hole is remarkably simple. The no-hair theorem tells us that a black hole is described by just two numbers: its mass M and spin angular momentum J a M 2 G/c, where a is the dimensionless spin parameter that is bounded to lie in the range 1 a 1 1. To understand the behavior of black holes, and ultimately to test general relativity in the strong field regime, it is essential that we establish reliable methods 1 In principle, a black hole can have a third parameter, electric charge, but this is unlikely to be important in astronomical settings. 1

16 CHAPTER 1. INTRODUCTION 2 of measuring their spins. The importance of measuring spin is widely recognized. For example, the science cases for two of the three large space missions considered in the US Decadal Survey, namely LISA and IXO, featured the measurement of black hole spin as a primary science goal. The measurement of spin is likewise a key objective for X-ray missions that are nearing launch, including Astro-H, GEMS, and ASTROSAT. Moreover, with the advent of Advanced LIGO, knowledge of black hole spin has become critical for calculating the expected gravitational waveforms from merging systems in which one or both objects are spinning black holes; measurements of spin are informing this work (Campanelli et al. 2006). Spin measurements are likewise being applied to problems in fundamental physics, e.g., Arvanitaki et al. (2010). Within astrophysics, spin data have provided the first observational evidence that relativistic jets can be powered directly by the spin energy of black holes (Narayan & McClintock 2012; Blandford & Znajek 1977). With modest improvements to current methodologies, it may soon be possible to test for violations of the no-hair theorem (see e.g., McClintock et al. 2011). Currently, two means of measuring spin have been developed and applied: the continuum-fitting (CF; Zhang et al. 1997) and Fe Kα (Fabian et al. 1989) methods. Each method has been used to estimate the spins of approximately ten stellar-mass black holes; the Fe Kα method has additionally yielded the spins of about ten supermassive black holes in active galactic nuclei (AGN). The foundation of both the CF and Fe Kα methods is the existence of an innermost stable circular orbit (ISCO) for particles orbiting a black hole. Outside

17 CHAPTER 1. INTRODUCTION 3 the ISCO, the gas slowly spirals inward through a series of nearly circular orbits. Inside the ISCO, there are no stable orbits for the accreting gas, and so it plunges into the black hole on a dynamical timescale ( 1ms for a stellar-mass black hole). As a result, the accretion disk is truncated at the ISCO. Therefore, by measuring the inner-disk radius, R in, we are equivalently determining the size of the ISCO radius, R ISCO. Since the dimensionless ISCO radius R ISCO /(GM/c 2 ), is a purely monotonic function of black hole spin, knowledge of R ISCO /M equivalently gives the value of a black hole s spin (Figure 1.1). As spin increases from a = 0 to a = 1, the dimensionless ISCO radius decreases sharply from 6 to 1 (e.g., Bardeen et al. 1972). This large change in radius over the range of possible spins is what enables us to securely measure spin using estimates of R ISCO /M. In the CF method, which is the focus of this thesis, one uses thermal continuum radiation from the accretion disk to measure R ISCO /M. In the principal alternative approach, the Fe Kα method, one also measures R ISCO /M, but using line emission from the disk instead. The line emission, which is fluoresced over the disk s surface, experiences a strong gravitational redshift at the inner reaches of the disk. As a result, the full breadth of the observed line profile can be used to determine the quantity R ISCO /M and thereby spin. The remainder of this chapter is organized as follows. In Section 1.1, we discuss the CF method in more detail. In Sections respectively we introduce the three central subjects of this dissertation: a new spectral model that has significantly extended the reach of the CF method; tests of two foundational assumptions of the

18 CHAPTER 1. INTRODUCTION R ISCO / M (G/c 2 ) a * Figure 1.1. The ISCO radius versus black hole spin. As spin changes from -1 to 0 to 1, the dimensionless ISCO radius R ISCO /(GM/c 2 ) decreases from 9 to 6 to 1. A horizontal line marks the ISCO radius for a = 0, R ISCO = 6GM/c 2. Notice that the relationship is monotonic and nonlinear and that the scaling of R ISCO /M with spin is steepest at the highest spin values.

19 CHAPTER 1. INTRODUCTION 5 method; and new CF measurements of spin for two black holes. 1.1 Continuum-Fitting Measurements of Spin In the CF method one measures R ISCO and determines spin by modeling the multitemperature thermal emission of the accretion disk. For the method to succeed, it is essential to have (1) a reliable theoretical model of the accretion disk; (2) accurate estimates of black hole mass M, disk inclination i, and source distance D; as well as (3) X-ray spectra that display a strong thermal component of emission. We discuss each of these three elements in turn. 1. The relativistic accretion disk model used is an elaborated and slightlycorrected version of the classic model of Novikov & Thorne (1973), which describes the thermal spectrum produced by a razor-thin accretion disk channeling gas onto a black hole. The model employed, which is referred to as kerrbb2 (Li et al. 2005; Davis & Hubeny 2006; McClintock et al. 2006), includes all relativistic effects and also includes the effects of limb-darkening, self-irradiation of the disk due to light bending, and the effects of spectral hardening. 2. Measuring R ISCO is analogous to measuring the radius of a star (with known distance) from its observed flux and temperature. In this analogy, X-ray flux and temperature determine the solid angle of the disk, from which R ISCO can be simply deduced if one knows D and the disk inclination i, which is usually assumed to be the same as the inclination of the orbital plane. Lastly, it is also

20 CHAPTER 1. INTRODUCTION 6 necessary to know M in order to obtain the dimensionless radius R ISCO /M, which is equivalent to knowing a (see Figure 1.1). The measurements of M, i and D are generally obtained by analyzing ground-based data obtained using optical and infrared instruments (e.g., Orosz et al. 2009). 3. Finally, one requires that the X-ray spectrum contain a strong thermal component. Usually, useful data is obtained in the thermal-dominant spectral state (Remillard & McClintock 2006), which is typified by soft disk emission and a relatively weak Compton power law. The only other strong requirement is that the disk must be well approximated by the razor-thin model employed. To achieve this, a luminosity restriction L < 0.3 L Edd is applied, where L Edd = (M/M ) erg s 1 is the Eddington limit (McClintock et al. 2006). Above this threshold, the disk scale-height flares beyond the thin disk regime and enters into a domain described by slim-disk models (e.g., S adowski et al. 2011). The CF method has so far been used to measure the spins of ten stellar-mass black holes. Those results are summarized in Table 1.1. Notably, all the spins are prograde (a > 0), and their values span the allowed range from 0 to Spectral Model The spectra of accreting stellar-mass black holes reveal a ubiquitous high-energy power-law component of emission (e.g., Remillard & McClintock 2006). This power law is generally attributed to the Compton up-scattering of thermal accretion-disk

21 CHAPTER 1. INTRODUCTION 7 Table 1.1. Continuum-Fitting Spin Measurements Black Hole a Reference M33 X ± 0.05 Liu et al. 2008, 2010 LMC X a Davis et al LMC X Gou et al A ± 0.19 Gou et al U a Shafee et al XTE J b 0.28 Chapter 7; Steiner et al XTE J a Shafee et al H ± 0.3 Chapter 6; Steiner et al GRS > 0.98 McClintock et al Cyg X 1 > 0.95 Gou et al Note. Errors are 1σ. a Value is provisional. b Using both CF and Fe Kα methods, the jointly-measured spin is a = 0.49.

22 CHAPTER 1. INTRODUCTION 8 emission by a hot ( 10 9 K) and optically thin outer atmosphere termed the corona. We adopt an empirical stance and develop a model for the effect of the corona on black hole spectra. Using the observed spectral shape of the component as a template, we assume that Compton-scattering occurs between thermal disk photons and coronal electrons. We assume that the system is generally symmetric with a scattering optical depth which is uniform and independent of the photon energy. In Chapter 2, we present the Comptonization model which results from these simple assumptions and implement a spectral fitting package for general use. In Chapter 3, we apply the Comptonization routine to thermal accretion disk spectra to achieve a composite model of Comptonized accretion-disk emission and investigate the implications of this unified Comptonized-disk model. We study a range of spectral states to test the scope of our model and to empirically assess its performance. The composite model developed in these chapters will be employed in all later applications of continuum fitting throughout this dissertation, and has been applied to estimate the spins of half of the black holes in Table Foundations The ISCO The single most critical assumption underpinning current measurements of black hole spin is the asserted link between spin and R in ; namely, the assumption that

23 CHAPTER 1. INTRODUCTION 9 the accretion disk terminates at the ISCO. Such a relationship naturally results from geometrically thin hydrodynamic accretion flows (Novikov & Thorne 1973; Shakura & Sunyaev 1973), but can in principle break down when the accretion disk is strongly magnetized or becomes geometrically thick. The foundational assumption that the ISCO corresponds to the disk s inner edge, and that R in therefore maps directly to spin is the basis for both the CF and Fe Kα methods of measuring spin. This assumption is supported by recent general relativistic, magnetohydrodynamic simulations (GRMHD; Shafee et al. 2008; Penna et al. 2010; Kulkarni et al. 2011; Noble et al. 2011). One consequence of the association of black hole spin and the ISCO radius is that in the thin-disk regime we consider, the inner-disk radius should be constant. It should not vary, e.g., with the brightness of the source. In Chapter 4, we explore the constancy of the inner radius empirically for the binary system LMC X 3, a persistent black hole which over the last three decades has been observed more regularly than nearly any other black hole source Spin-Orbit Alignment The second crucial assumption of the CF method is that a black hole s spin angular momentum is aligned with the orbital angular momentum of the binary system. This expectation is grounded in knowledge that accretion torques acting over millions of years can readily cause alignment in stellar-mass black hole binary systems (e.g., Martin et al. 2008). However, while theoretically motivated, this supposed alignment has yet to be rigorously tested.

24 CHAPTER 1. INTRODUCTION 10 The black hole binary XTE J (hereafter J1550) provides a unique opportunity to make such a test. J1550 is a poster-child microquasar system which underwent a violent outburst in 1998 followed by an atypical re-ignition and subsequent decay. During its outburst, J1550 launched a pair of superluminal jets. These jets were observed several years later in X-rays by Chandra, shocking against the ambient interstellar medium (Corbel et al. 2002), the first discovery of its kind. By tracking the motion of the nearly pc-scale jets along the plane of the sky, in Chapter 5, we fit for the jet positions and solve for the jet inclination angle, which is presumed to match the angle of the spin-axis of J1550 s black hole. Meanwhile, the binary inclination angle of the system has been previously measured by our group using optical and infrared light-curve and radial-velocity data (Orosz et al. 2011). We combine these two measurements and produce the first strong test for alignment along the line-of-sight. 1.4 New Spin Measurements H The microquasar H (hereafter H1743), like J1550, also produced large-scale X-ray and radio jets. In Chapter 6, we analyze the motion of these jets on the plane of the sky (just as we did for J1550) and thereby determine the distance to H1743 and the inclination of the black hole s spin axis. Combining these measurements of D and i with our knowledge of the mass distribution for black holes in transient systems, we obtained an estimate of H1743 s spin. This is the first time that the CF

25 CHAPTER 1. INTRODUCTION 11 method has been used to measure the spin of a black hole despite the absence of any dynamical constraints on the parameters D, i, and M even the orbital period of H1743 is unknown XTE J : A Joint CF and Fe Kα Study J1550 is significant for having produced the first and most dramatic example of X-ray jets in a black hole microquasar, and for also having produced a resonant 3:2 pair of high-frequency quasi-periodic oscillations (QPOs; Remillard et al. 2002a; Sobczak et al. 2000b). These QPOs are thought to be produced in the innermost regions of the accretion disk and their frequencies are widely believed to depend on only the mass and spin of the black hole (e.g., Remillard & McClintock 2006, and references therein). Because of the importance of J1550, we performed ground-based optical observations and derived new precise estimates for the binary parameters and distance (M = 9.1 ± 0.6 M, i = 74.7 ± 3.8 degrees, D = 4.4 ± 0.5 kpc; Orosz et al. 2011). Building on these results, in Chapter 7, we determine J1550 s spin via the CF method using 50 RXTE spectra, and pay strict attention to all known sources of systematic error. At the same time, in an effort to improve this CF measurement and to check the cross-consistency with the Fe Kα method, we worked in collaboration with Fe Kα experts to obtain an independent measurement of J1550 s spin from the premier models of spectral reflection in black hole binaries. In Chapter 7, both CF and Fe Kα results are presented in turn; ultimately, we combine the two measurements to

26 CHAPTER 1. INTRODUCTION 12 produce a combined estimate of J1550 s spin.

27 Chapter 2 A Simple Comptonization Model J. F. Steiner, R. Narayan, J. E. McClintock, & K. Ebisawa Publications of the Astronomical Society of the Pacific, Vol. 121, pp , 2009 Abstract We present an empirical model of Comptonization for fitting the spectra of X-ray binaries. This model, named simpl, has been developed as a package implemented in XSPEC. With only two free parameters, simpl is competitive as the simplest model of Compton scattering. Unlike the pervasive standard power-law model, simpl incorporates the basic features of Compton scattering of soft photons by energetic coronal electrons. Using a simulated spectrum, we demonstrate that simpl closely matches the behavior of physical Comptonization models which consider the effects of optical depth, coronal electron temperature, and geometry. We present fits to RXTE spectra of the black-hole transient H and a BeppoSAX spectrum of 13

28 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 14 LMC X 3 using both simpl and the standard power-law model. A comparison of the results shows that simpl gives equally good fits, while eliminating the troublesome divergence of the standard power-law model at low energies. simpl is completely flexible and can be used self-consistently with any seed spectrum of photons. We show an example of how simpl unlike the standard power law teamed up with diskbb (the standard model of disk accretion) provides a uniform disk normalization that is unaffected by moderate Comptonization. 2.1 Introduction Spectra of X-ray binaries typically consist of a soft (often blackbody or bremsstrahlung) component and a higher-energy tail component of emission, which we refer to generically as a power law throughout this work. The origin of the power-law component in both neutron-star and black-hole systems is widely attributed to Compton up-scattering of soft photons by coronal electrons (White et al. 1995; Remillard & McClintock 2006, hereafter RM06). While this interpretation is not unique, in this work, we adopt the prevailing view that Compton scattering is the mechanism that generates the observed power law. This component is present in the spectra of essentially all X-ray binaries, and it occurs for a wide range of physical conditions. The tail emission is generally modeled by adding a simple power-law component to the spectrum, e.g., via the model powerlaw in the widely used fitting package XSPEC (Arnaud 1996). A few of the many applications where power-law models are employed include: modeling the thermal continuum (Narayan et al. 2008) or the

29 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 15 relativistically-broadened Fe K line (Miller et al. 2008b) in order to obtain estimates of black-hole spin; modeling the surrounding environment of compact X-ray sources, such as a tenuous accretion-disk corona (White & Holt 1982) or a substantial corona that scatters photons up to MeV energies (Gierliński et al. 1999); and classifying patterns of distinct X-ray states, e.g., in black-hole binaries (RM06). Because of the importance of the power-law component, several physical models have been developed to infer the conditions of the hot plasma that causes the Comptonization. Models of this variety that are available in XSPEC are comptt (Titarchuk 1994), eqpair (Coppi 1999), comptb (Farinelli et al. 2008), bmc (Titarchuk et al. 1997), compbb (Nishimura et al. 1986), thcomp (Życki et al. 1999), compls (Lamb & Sanford 1979), and compps (Poutanen & Svensson 1996). It is essential to use such physical models when one is focused on understanding the physical conditions and structure of a scattering corona or other Comptonizing plasma. Often, however, the physical conditions of the Comptonizing medium are poorly understood or are not of interest, and one is satisfied with an empirical model that seeks to match the data with no pretense that the model can sufficiently discern the underlying physics. The model powerlaw is one such empirical model which has been extraordinarily widely used in modeling black-hole and neutron-star binaries (see text & references in White et al. 1995; Tanaka & Lewin 1995; Brenneman & Reynolds 2006; RM06) and AGN (e.g., Zdziarski et al. 2002; Brenneman & Reynolds 2006). However, powerlaw introduces a serious flaw: at low energies it rises without limit. The divergence at low energies, which is not expected for Comptonization, significantly corrupts the parameters returned by the model

30 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 16 component with which it is teamed (e.g., the widely used disk blackbody component diskbb; Section 2.3). An excellent alternative to the standard power-law model for describing Compton scattering is given by convolution using a scattering Green s function, formulated decades ago (Shapiro et al. 1976; Rybicki & Lightman 1979; Sunyaev & Titarchuk 1980; Titarchuk 1994). In this approach the power-law is generated selfconsistently via Compton up-scattering of a seed photon distribution; consequently, the power-law naturally truncates itself as the seed distribution falls off at low energies. In this chapter, we present our implementation of a flexible convolution model named simpl that can be used with any spectrum of seed photons. For a Planck distribution we show that simpl gives identical results to bmc, as expected since the two models are functionally equivalent (Section 2.2.5). Although simpl has only two free parameters, the same number as the standard powerlaw, this empirical model is nevertheless able to very successfully fit data simulated using comptt, a prevalent physical model of Comptonization (Section 2.2.4). We analyze data for two black-hole binaries and illustrate the flexibility of simpl by convolving simpl with diskbb, the workhorse accretion disk model which has been used for decades (Mitsuda et al. 1984). The principal result is that simpl in tandem with diskbb enables one to obtain values for the disk-normalization parameter for more heavily Comptonized data that are consistent with those found for weakly-comptonized data (see Section 2.3.2). The standard power law, on the other hand, delivers very inconsistent normalization values. This is shown in greater detail in (Steiner et al. 2009a).

31 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 17 In Section 2.2 we outline the model and in Section 2.3 we present a case study with several examples. We discuss a prospective application of the model in Section 2.4 and conclude with a summary in Section The Model: simpl The model simpl (SIMple Power Law) functions as a convolution that converts a fraction of input seed photons into a power law (see eq. [2.1]). The model is currently available in XSPEC 1. In addition to simpl-2, which is our implementation of the classical model described by Shapiro et al. (1976) and Sunyaev & Titarchuk (1980), which corresponds to both up- and down-scattering of photons, we offer an alternative bare-bones implementation in which photons are only up-scattered in energy. The physical motivations behind the two versions of the model are described in Section 2.2.1, and the corresponding scattering kernels the Green s functions are given in equation (2.3) and equation (2.4), respectively. The parameters of simpl and the standard powerlaw model are similar. Their principal parameter, the photon index Γ, is identical. However, in the case of simpl, the normalization factor is the scattered fraction f SC, rather than the photon flux. The goal of simpl is to characterize the effects of Comptonization as simply and generally as possible. In this spirit, all details of the Comptonizing medium, such as its geometry (slab vs. sphere) or physical characteristics (optical depth, temperature, thermal vs. non-thermal electrons), which would require additional 1 see

32 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 18 parameters for their description, are omitted. It is appropriate to employ simpl when the physical conditions of the Comptonizing medium are poorly understood or are not of interest. When the details of the Comptonizing medium are known, or are the main object of study, one should obviously use other models (e.g., comptt, compps, thcomp, etc.), which are designed specifically for such work. simpl, on the other hand, is meant for those situations in which a Compton power-law component is present in the spectral data and needs to be included in the model but is not the primary focus of interest. simpl should thus be viewed as a broad-brush model with the same utility as powerlaw but designed specifically for situations involving Comptonization. By virtue of being a convolution model, simpl mimics physical reprocessing by tying the power-law component directly to the energy distribution of the input photons. The most important feature of the model is that it produces a power-law tail at energies larger than the characteristic energy of the input photons, and that the power law does not extend to lower energies. This is precisely what one expects any Compton-scattering model to do and is a general feature of all the physical Comptonization models mentioned above. In contrast, the model powerlaw simply adds to the spectrum a pure power-law component that reaches all the way downward to arbitrarily low energies. The difference between simpl and powerlaw is thus most obvious at soft X-ray bands where simpl cuts off in a physically natural way, as appropriate for Comptonization, whereas powerlaw continues to rise without limit (e.g., see Yao et al. 2005). Two assumptions underlie simpl. The first is that all soft photons have the

33 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 19 same probability of being scattered (e.g., the Comptonizing electrons are distributed spatially uniformly). This is a reasonable assumption when one considers that, even in the best of circumstances, almost nothing is known about the basic geometry of the corona. For example, usually the corona is variously and crudely depicted as a sphere, a slab, or a lamp post. The second assumption is that the scattering itself is energy independent. This is again reasonable given the soft thermal spectra of the seed photons that are observed for black-hole and neutron-star accretion disks, with typical temperatures of 1 kev and a few kev, respectively. For example, in the extreme case of a 180 back-scatter off a stationary electron, a 3 kev seed photon suffers only a 1% loss of energy, and even a 10 kev photon loses only 4% of its initial energy.

34 o CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 20 ν F ν SIMPL 1 SIMPL 2 100% 50% 25% 10% 1% seed Energy (kev) Figure 2.1. Spectral energy density vs. photon energy for a sample spectrum calculated with simpl-1 (solid lines) and simpl-2 (dashed lines). The models conserve photons and Comptonize a seed spectrum, which in the case shown is diskbb with kt = 1 kev (black line). Ascending colored lines show increasing levels of scattering, from f SC = 1 100%.

35 Table 2.1. Results of Fitting a Simulated comptt Spectrum MODEL χ 2 ν/ν N H Γ f SC Norm(PL) a kt 0 Norm b kt e τ c (10 22 cm 2 ) (kev) (kev) comptt c simpl-1 bb 1.00/ ± ± ± ± ± 0.4 simpl-2 bb 1.06/ ± ± ± ± ± 0.3 compbb 1.05/ ± ± ± ± ± 0.03 bb+powerlaw 2.02/ ± ± 0.01 (5.0 ± 0.2) ± ± 0.02 a powerlaw normalization given at 1 kev in photons s 1 cm 2 kev 1. b bb and compbb normalization = R/km D/10 kpc 2 for a blackbody of radius R at a distance D; comptt normalization is undefined. c comptt model set to disk geometry (geometry switch = 1). CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 21

36 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 22 Figure 2.1 shows sample outputs from simpl when the input soft photons are modeled by the multi-temperature disk blackbody model diskbb (Mitsuda et al. 1984). Results are shown for both simpl-2 and simpl-1, our alternative version of simpl that includes only up-scattering of photons; the spectra are shown for Γ = 2.5 and a range of values of f SC. Note the power-law tails in the model spectra at energies above the peak of the soft thermal input and the absence of an equivalent power-law component at lower energies. This is the primary distinction between simpl and powerlaw. simpl-2 and simpl-1 give similar spectra, but the spectrum from simpl-1 has a somewhat stronger power-law tail for the same value of f SC. This is because simpl-1 transfers all the scattered photons to the high energy tail, whereas simpl-2 has double-sided scattering. Therefore, for the same value of f SC, fewer photons are scattered into the high-energy tail with simpl-2. Correspondingly, when fitting the same data, simpl-2 returns a larger value of f SC compared to simpl-1 (for examples, see Section 2.3 and Table 2.2) Green s Functions Given an input distribution of photons n in (E 0 )de 0 as a function of photon energy E 0, simpl computes the output distribution n out (E)dE via the integral transform: [ Emax ] n out (E)dE = (1 f SC )n in (E)dE + f SC n in (E 0 )G(E; E 0 )de 0 de. (2.1) E min A fraction (1 f SC ) of the input photons remains unscattered (the first term on the right), and a fraction f SC is scattered (the second term). Here, E min and E max are the minimum and maximum photon energies present in the input distribution, and G(E; E 0 ) is the energy distribution of scattered photons for a δ-function input at

37 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 23 energy E 0, i.e., G(E; E 0 ) is the Green s function describing the scattering. Equation (2.1) assumes that every photon ultimately escapes to infinity, either unscattered (the first term on the right-hand side) or after Compton scattering (the second term). In the context of a disk-corona model we note that as much as half the scattered photons (the exact fraction depends on geometry) are redirected towards the disk. Computing the fate of these photons is the goal of sophisticated reflection models, e.g., reflionx (Ross & Fabian 2005) and PEXRAV (Magdziarz & Zdziarski 1995). Equation (2.1) ignores all these details and simply assumes that all the photons that return to the disk are effectively scattered from the surface with no change in energy. In the opposite extreme, we may wish to assume that all the returning photons are fully absorbed and thermalized. In this limit, we would replace equation (2.1) with: [ Emax ] n out (E)dE = (1 f SC )n in (E)dE + (f SC /2) n in (E 0 )G(E; E 0 )de 0 de. (2.2) E min Clearly, the real situation is somewhere in between (2.1) and (2.2). The version of simpl currently implemented in XSPEC makes use of equation (2.1), though it would be straightforward to change it to equation (2.2). We now describe the specific prescriptions we use for simpl-2 and simpl-1. We also discuss the physical motivations behind these prescriptions, drawing heavily on the theory of Comptonization as described by Rybicki & Lightman (1979, hereafter RL79).

38 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL simpl-2: In sec. 7.7, RL79 discuss the case of unsaturated repeated scattering by nonrelativistic thermal electrons. Following Shapiro, Lightman, & Eardley (1976), they solve the Kompaneets equation and show that Comptonization produces a power-law distribution of photon energies (eq. 7.76d in RL79). There are two solutions for the photon index Γ: Γ 1 = y, Γ 2 = y, where the Compton y parameter is given by y = (4kT e /m e c 2 )Max(τ es, τes 2 ). Here, kt e is the electron temperature and τ es is the optical depth to electron scattering. Up-scattered photons have a power-law energy distribution with photon index Γ 1 and down-scattered photons have a different power-law distribution with photon index Γ 2. We model this case of nonrelativistic electrons with the following Green s function (Sunyaev & Titarchuk 1980; Titarchuk 1994; Ebisawa 1999), which corresponds to the model simpl-2: G(E; E 0 )de = (Γ 1)(Γ + 2) (1 + 2Γ) (E/E 0 ) Γ de/e 0, E E 0 (E/E 0 ) Γ+1 de/e 0, E < E 0. (2.3) The function is continuous at E = E 0, is normalized such that it conserves photons, and holds for all Γ > 1. Substituting (2.3) in (2.1) we see that simpl-2 has two parameters: f SC and Γ. Note that although the model makes use of two power laws, their slopes are not independent.

39 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 25 As in the case of the standard power law, simpl includes no high energy cutoff. Technically, for any complete model of Comptonization, the up-scattered power-law distribution is cut off for photon energies larger than kt e. To avoid increasing the complexity of our model, we have ignored this detail; extra parameters could easily be added to account for high energy attenuation if desired. By keeping the model very basic, simpl is a direct two-parameter replacement for the standard power law while bridging the divide between the latter model and physical Comptonization models simpl-1 The Green s function (2.3) is obtained by solving the Kompaneets equation, which assumes that the change in energy of a photon in a single scattering is small. This assumption is not valid when the Comptonizing electrons are relativistic. In sec. 7.3 of their text, RL79 discuss Compton scattering by relativistic electrons with a power-law distribution of energy: n e (E e )de e Ee p de e. In the limit when the optical depth is low enough that we only need to consider single scattering, they show that the Comptonized spectral energy distribution (SED) is a power law of the form P(E)dE E (p 1)/2. Equivalently, the photon energy distribution takes the form n(e)de E Γ, with a photon index Γ = (p + 1)/2. Hardly any photons are down-scattered in energy. In sec. 7.5, RL79 show that repeated scatterings produce a power-law SED even when the relativistic electrons have a non-power-law distribution (see also Titarchuk & Lyubarskij 1995). In terms of τ es and the mean amplification of photon

40 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 26 energy per scattering A, the Comptonized photon energy distribution takes the form n(e)de E Γ with a photon index Γ = 1 ln τ es / lna. For the specific case of a thermal distribution of electrons with a relativistic temperature kt e m e c 2, the amplification factor is given by A = 16(kT e /m e c 2 ) 2. Once again, hardly any photons are down-scattered. For both cases discussed above, Comptonization is dominated by up-scattering and produces a nearly one-sided power-law distribution of photon energies. This motivates the following Green s function, valid for Γ > 1, which we refer to as the model simpl-1: (Γ 1)(E/E 0 ) Γ de/e 0, E E 0 G(E; E 0 )de = 0, E < E 0. (2.4) The normalization factor (Γ 1) ensures that we conserve photons. Although simpl-1 is most relevant for relativistic Comptonization, it can also be used as a stripped-down version of simpl-2 for non-relativistic coronae. The reason is that the low-energy power-law (E/E 0 ) Γ+1 in equation (2.3) almost never has an important role. There is not much power in this component, and what little contribution it makes is indistinguishable from the input soft spectrum. Therefore, even for the case of nonrelativistic thermal Comptonization, for which the Green s function (2.3) is designed, there would be little difference if one were to use simpl-1 instead of simpl-2.

41 CHAPTER 2. A SIMPLE COMPTONIZATION MODEL Comparison to comptt To illustrate the performance of simpl relative to other Comptonization models, we have simulated a count BeppoSAX (Boella et al. 1997) observation using the comptt model in XSPEC v12.4.0x. For our source spectrum, we adopt disk geometry, a Wien distribution of seed photons at kt 0 = 1 kev, and a hydrogen column density of N H = cm 2. We set the optical depth and temperature of the Comptonizing medium to τ c = 2 and kt e = 40 kev. Our simulation uses the LECS, MECS, and PDS detectors on BeppoSAX, which span a wide energy range kev (for details on the instruments, see Section 2.3). The total number of counts in the simulated spectra ( ) corresponds to a 3 ks observation of a 1 Crab source.

42 Table 2.2. Spectral Fit Results phabs simpl diskbb powerlaw Source Mission MJD χ 2 ν/ν L disk a L Edd N H Ver. b Γ f SC kt Norm c Γ Norm(PL) d State Detector (10 22 cm 2 ) (kev) H1743 RXTE / ± ± ± ± ± 0.42 SPL PCA 1.11/ ± 0.10 S ± ± ± ± / ± 0.11 S ± ± ± ± 33 H1743 RXTE / ± ± ± ± ± 0.04 TD PCA 0.93/ ± 0.10 S ± ± ± ± / ± 0.10 S ± ± ± ± 25 LMC X 3 BeppoSAX e / ± ± ± ± ± TD LECS,MECS, 1.08/ ± S ± ± ± ± 1.2 PDS 1.08/ ± S ± ± ± ± 1.1 a Bolometric ( kev) luminosity of the disk component in Eddington units. For H1743, we adopt nominal values: M = 10 M, D = 9.5 kpc, and i = 60. The fiducial values used for LMC X 3 are M = 7.5 M and i = 67 (Cowley et al. 1983; Orosz 2003). For fits using simpl, this quantity describes the seed spectral luminosity. b Version of simpl being used, i.e., S1 for simpl-1 and S2 for simpl-2. c For an accretion disk inclined by i to the line of sight, with inner radius R in at distance D, Norm = Rin /km D/10 kpc 2 cos i. d powerlaw normalization given at 1 kev in photons s 1 cm 2 kev 1. e The cross-normalizations for C LM LECS/MECS and C PM PDS/MECS are fitted from and respectively. C LM = ± 0.283, ± 0.008, ± for the fits with powerlaw, simpl-1, and simpl-2. C PM is pegged at 0.93 for the same fits. Note. All errors are presumed Gaussian and quoted at 1σ. CHAPTER 2. A SIMPLE COMPTONIZATION MODEL 28